Question: A circle with radius $2$ has a sector with a $225^\circ$ central angle. What is the area of the sector? ${4\pi}$ $\color{#9D38BD}{225^\circ}$ ${\dfrac{5}{2}\pi}$ ${2}$
Answer: First, calculate the area of the whole circle. Then the area of the sector is some fraction of the whole circle's area. $A_c = \pi r^2$ $A_c = \pi (2)^2$ $A_c = 4\pi$ The ratio between the sector's central angle $\theta$ and $360^\circ$ is equal to the ratio between the sector's area, $A_s$ , and the whole circle's area, $A_c$ $\dfrac{\theta}{360^\circ} = \dfrac{A_s}{A_c}$ $\dfrac{225^\circ}{360^\circ} = \dfrac{A_s}{4\pi}$ $\dfrac{5}{8} = \dfrac{A_s}{4\pi}$ $\dfrac{5}{8} \times 4\pi = A_s$ $\dfrac{5}{2}\pi = A_s$